Analysis of the upper bounds for the integer ambiguity validation statistics

Integer ambiguity validation is an essential quality control step for high-precision positioning and navigation with global navigation satellite systems (GNSS). In order to validate the resolved integer ambiguities, statistical tests, such as the R-ratio test, F-ratio test, W-ratio test, difference test, and projector test, have been favored. In practice, the critical values for these statistical tests are determined either empirically or from the assumed distributions. However, previous research has revealed that some of these statistics have upper bounds, which can be obtained from simulations. In this contribution, we find that under the framework of the integer aperture estimation, the upper bounds for these ambiguity validation statistical tests can be derived without actual measurements or simulation. As a result, the assumed distributions for these statistical tests are inappropriate. According to the derivation, it has been concluded that the upper bounds of these ambiguity validation tests depend only on the ambiguity geometry (e.g., the float ambiguity variance–covariance matrix) and can be obtained at the design stage of GNSS positioning. Thus, the critical value for these ambiguity validation statistics has a rigorous range, and it should be chosen to be smaller than a priori derived upper bound. Otherwise, no integer ambiguities can be obtained.